Formally proving $\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$?

Question

$\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$

This fact seems pretty obvious but how would I formally prove it, is there a painless way?

Answer 1

$$\bigcup_{k=1}^{\infty}\left\{-k<X\leqslant-k+1\right\}=\left\{X\leqslant0\right\}$$